Abstract

This paper extends Burckhardt’s solution for diffraction from a thick grating to include complex dielectric constants and nonsinusoidal stratifications. This allows any realistic periodic structure to be handled. Computed results are compared with coupled-wave theory, as described by Kogelnik, with emphasis on strongly absorbing gratings such as those made by photographing an interference pattern. Finally, some experimental holographic data are compared with computations that take into account the photographic nonlinearity between exposure and dielectric constant.

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  1. F. G. Kaspar and R. L. Lamberts, J. Opt. Soc. Am. 58, 970 (1968).
  2. F. G. Kaspar, R. L. Lamberts, and C. D. Edgett, J. Opt. Soc. Am. 53, 1289 (1968).
  3. K. Biedermann, Optik 28, 160 (1968).
  4. A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
  5. J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).
  6. R. L. Powell and Karl A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
  7. J. S. Zelenka and J. R. Varner, Appl. Opt. 7, 2107 (1968).
  8. C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).
  9. H. Kogelnik, Bell System Tech. J. 48, 2909 (1969).
  10. D. Kermisch, Ph.D. thesis, Polytechnic Institute of Brooklyn (1968).
  11. E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, 1958), p. 413.
  12. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. P., Oxford, 1965), p. 24.
  13. R. L. Sanderson and W. Streifer, Appl. Opt. 8, 131 (1969).
  14. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 613.
  15. R. L. Lamberts, J. Opt. Soc. Am. 60, 1389 (1970).
  16. The relation between εi and exposure as expressed in Eq. (15) is assumed valid even when exposure is a function of position [recall that Eq. (15) was derived for a homogeneous medium] although, admittedly, transmittance as such is not well defined operationally for high-frequency patterns in thick emulsions. The problem arises from the difficulty of measuring a unique value of transmittance for high-frequency patterns in thick films. Different methods of illumination result in different measured values of T because of light scatter. At high spatial frequencies, Eq. (13) is no longer valid, but this does not invalidate the functional relation between εi and exposure as derived from Eq. (15) and the D-log(exposure) curve. The quantity εis is uniquely defined; so long as its relation to exposure is independent of spatial frequency, then this relation as derived by use of Eq. (15) is correct.
  17. E. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).
  18. H. M. Smith, J. Opt. Soc. Am. 62, 802 (1972).
  19. H. Kogelnik, J. Opt. Soc. Am. 57, 431 (1967).
  20. I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Integral No. 3.915-4, p. 482.

Biedermann, K.

K. Biedermann, Optik 28, 160 (1968).

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 613.

Burckhardt, C. B.

C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).

Edgett, C. D.

F. G. Kaspar, R. L. Lamberts, and C. D. Edgett, J. Opt. Soc. Am. 53, 1289 (1968).

Goodman, J. W.

J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).

Gradshteyn, I.

I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Integral No. 3.915-4, p. 482.

Kaspar, F. G.

F. G. Kaspar and R. L. Lamberts, J. Opt. Soc. Am. 58, 970 (1968).

F. G. Kaspar, R. L. Lamberts, and C. D. Edgett, J. Opt. Soc. Am. 53, 1289 (1968).

Kermisch, D.

D. Kermisch, Ph.D. thesis, Polytechnic Institute of Brooklyn (1968).

Knight, G. R.

J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).

Kogelnik, H.

H. Kogelnik, Bell System Tech. J. 48, 2909 (1969).

H. Kogelnik, J. Opt. Soc. Am. 57, 431 (1967).

Kozma, A.

E. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).

A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).

Lamberts, R. L.

F. G. Kaspar, R. L. Lamberts, and C. D. Edgett, J. Opt. Soc. Am. 53, 1289 (1968).

F. G. Kaspar and R. L. Lamberts, J. Opt. Soc. Am. 58, 970 (1968).

R. L. Lamberts, J. Opt. Soc. Am. 60, 1389 (1970).

Leith, E.

E. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).

Marks, J.

E. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).

Massey, N.

E. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).

Powell, R. L.

R. L. Powell and Karl A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).

Ryzhik, I.

I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Integral No. 3.915-4, p. 482.

Sanderson, R. L.

R. L. Sanderson and W. Streifer, Appl. Opt. 8, 131 (1969).

Smith, H. M.

H. M. Smith, J. Opt. Soc. Am. 62, 802 (1972).

Stetson, Karl A.

R. L. Powell and Karl A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).

Streifer, W.

R. L. Sanderson and W. Streifer, Appl. Opt. 8, 131 (1969).

Upatnieks, J.

E. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).

Varner, J. R.

J. S. Zelenka and J. R. Varner, Appl. Opt. 7, 2107 (1968).

Watson, G.

E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, 1958), p. 413.

Whittaker, E.

E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, 1958), p. 413.

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. P., Oxford, 1965), p. 24.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 613.

Zelenka, J. S.

J. S. Zelenka and J. R. Varner, Appl. Opt. 7, 2107 (1968).

Other (20)

F. G. Kaspar and R. L. Lamberts, J. Opt. Soc. Am. 58, 970 (1968).

F. G. Kaspar, R. L. Lamberts, and C. D. Edgett, J. Opt. Soc. Am. 53, 1289 (1968).

K. Biedermann, Optik 28, 160 (1968).

A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).

J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).

R. L. Powell and Karl A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).

J. S. Zelenka and J. R. Varner, Appl. Opt. 7, 2107 (1968).

C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).

H. Kogelnik, Bell System Tech. J. 48, 2909 (1969).

D. Kermisch, Ph.D. thesis, Polytechnic Institute of Brooklyn (1968).

E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, 1958), p. 413.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. P., Oxford, 1965), p. 24.

R. L. Sanderson and W. Streifer, Appl. Opt. 8, 131 (1969).

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 613.

R. L. Lamberts, J. Opt. Soc. Am. 60, 1389 (1970).

The relation between εi and exposure as expressed in Eq. (15) is assumed valid even when exposure is a function of position [recall that Eq. (15) was derived for a homogeneous medium] although, admittedly, transmittance as such is not well defined operationally for high-frequency patterns in thick emulsions. The problem arises from the difficulty of measuring a unique value of transmittance for high-frequency patterns in thick films. Different methods of illumination result in different measured values of T because of light scatter. At high spatial frequencies, Eq. (13) is no longer valid, but this does not invalidate the functional relation between εi and exposure as derived from Eq. (15) and the D-log(exposure) curve. The quantity εis is uniquely defined; so long as its relation to exposure is independent of spatial frequency, then this relation as derived by use of Eq. (15) is correct.

E. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).

H. M. Smith, J. Opt. Soc. Am. 62, 802 (1972).

H. Kogelnik, J. Opt. Soc. Am. 57, 431 (1967).

I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Integral No. 3.915-4, p. 482.

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